We however stress that if \(V\in C^1({\mathbb {R}})\cap L^\infty ({\mathbb {R}})\)  is repulsive, i.e., \(xV^{\prime }(x)\le 0\) , then \(H\)  has no eigenvalues (see [1]}). Note that such a criterion also works for the general higher-order elliptic operator \(P(D)+V\)  in any dimensions \(n\ge 1\) .
Besides, we also notice that for a general selfadjoint operator \(\mathcal {H}\)  on \(L^{2}(\mathbb {R}^n)\) , even if \(\mathcal {H}\)  has a simple embedded eigenvalue, Costin–Soffer in [2]} have proved that \(\mathcal {H}+{\varepsilon }W\)  can kick off the eigenvalue located in a small interval under certain small perturbation of the potential \({\varepsilon }W\) .
